Using this argument, we can associate a 2D gap topological index (Chern number) to a gap of a 1D quasiperiodic model. In other words, using the quasiperiodic order, we quantize/flatten the Chern density such that it serves as a topological index for the 1D chain. This construction locks the Diophantine/Streda relation described above, such that at a given filling factor, when a gap occurs, an associated single-valued Chern density appears that corresponds to the Chern number of the related topological pump divided by \(2\pi \) . This discussion is transferable to other quasicrystals that have a Hamiltonian that is periodic with a phase shift \(\phi \) , which can be compensated by a translation, including in higher-dimensions [1]}.
